Pythagoras for a Tetrahedron
Why use this problem?
The problem introduces an attractive generalisation of Pythagoras' theorem to 3D.
There is the opportunity to explore different solution techniques. Some methods include:
- Using the area of a triangle formulae and Pythagoras's theorem.
- Using Heron's formula.
- Using vectors.
Key questions
What information is given?
In what different ways could we find the area of a triangle?
What can be deduced from the geometrical properties?
What are the unknowns and what notation shall we use?
What equations can we write down for the areas of the four triangular faces of the tetrahedron?
What equations can we write down involving the lengths of the edges of the tetrahedron?
How can we put all this together to get the required result?
Possible support
Try the problem Rectangular Pyramids.
Possible extension
Try using different methods to solve the problem.
Generalise the result to a cosine rule for a tetrahedron.
STEP Support Programme Foundation Assignment 5 asks a different question about the same tetrahedron.
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